A Cooperative Game-Based Sizing and Configuration of Community-Shared Energy Storage

Abstract

Sizing and configuring community-shared energy storage according to the actual demand of community users is important for the development of user-side energy storage. To solve this problem, this paper first proposes a community energy storage cooperative sharing mode containing multiple transaction types and then establishes a sizing and configuration model of community-shared energy storage based on a cooperative game among community users and energy storage operators, in which the loss caused by the capacity decay of energy storage is quantified by a dynamic power loss cost factor. To improve the solving efficiency, a distributed and cooperating solving method based on ADMM is used to solve the sizing and configuration model. On this basis, the bilateral Shapley method is used to allocate the total annual cost according to the marginal expected cost brought by each user. Compared with existing strategies, this paper calculates the economic benefits of community-shared energy storage based on several typical days of each year and quantifies the capacity decay of energy storage by a dynamic power loss cost factor which increases year by year to be closer to the real situation. Finally, the simulation verifies that the model proposed in this paper can be used for the sizing and configuration of community-shared energy storage. Compared with the original annual cost, the total annual cost of the community is reduced by 3.92%, and the annual operation cost of the community which equals annual electricity purchasing cost minus annual electricity selling income plus annual power loss cost is reduced by 25.6%.

1. Introduction

1.1. Background

As an important flexibility source [1], the deployment of energy storage is an important way to improve the flexibility of a power system [2] or an energy hub [3], which is also one of the reasons why energy storage receives more attention than before. The global energy consumption structure is transiting rapidly [4]. In recent years, with the increasing user-side electricity demand and the installed capacity of intermittent distributed energy resources [5], user-side energy storage is playing an increasingly important role in the grid [6]. However, the high investment cost of user-side energy storage and the low feed-in tariff severely limit the development of user-side energy storage [7,8]. If shared energy storage can be configured in the community and multiplexed among community users, it is expected to share the investment cost of energy storage, improve the utilization rate of energy storage, and reduce the electricity cost for community users, achieving multi-party win-win [9,10]. One of the most important issues is how to configure the capacity of community-shared energy storage (CSES) according to the actual demands of community users [11,12]. However, the existing studies are still inadequate for calculating the economic benefits during the life cycle of CSES, which leads to bias in the calculation of the CSES optimal configuration capacity. Therefore, the economic benefits during the life cycle of CSES need to be fully considered in the study of the sizing and configuration of CSES to maximize its economic benefits and thus promote the development of user-side energy storage.

1.2. Literature Review

At present, scholars at home and abroad have conducted much research on the problem of sizing and configuration of CSES, and one of the key points is the calculation method of economic benefits during the life cycle of CSES. Based on the electricity demand and PV power in a single typical day of the community, Lu et al. [13] configured the capacity of energy storage to minimize the comprehensive community cost and the waste rate of clean energy. Further, Edward Barbour et al. [14] compared the energy storage’s configuration capacity of different modes based on calculations with real power data over a month and concluded that the CSES mode can reduce the configuration capacity of energy storage. Cheng et al. [15] used the scenario analysis method to configure the capacity of CSES based on several typical day scenarios of a year.

In addition, since the performance of the energy storage system is constantly changing during the long period of operation, how quantifying the capacity decay of CSES during operation is also a key point in calculating the economic benefits of CSES. Cheng et al. [15] used a capacity loss factor to quantify the capacity decay of the energy storage system for one day. Further, Xie et al. [16] established a capacity decay model during the life cycle of the energy storage system by the rain flow method, which can accurately quantify the capacity change in the energy storage system each year. Chen et al. [17] used the power loss cost factor to quantify the economic loss caused by the capacity decay of the energy storage system.

Another problem in calculating the economic benefits during the life cycle of CSES is the increased computing scale and model-solving difficulty. For this problem, Shuai et al. [18,19] used the heuristic algorithm combined with the CPLEX solver to optimize the capacity configuration of shared energy storage and reduced unnecessary investment costs. Xie et al. [16] used an iterative method to optimize the capacity configuration of shared energy storage to quantify the impact of energy storage capacity decay on the configuration capacity, and the constructed two-layer model needs to be solved in each iteration.

For the calculation methods of economic benefits during the life cycle of CSES, the configuration results based on data in a single typical day [13] are highly dependent on the selection of the typical day, which leads to low reliability. Additionally, calculation methods based on data in only one month or one year [14,15] are still one-sided because the calculation of economic benefits during the life cycle of CSES is a long-timescale problem spanning many years. Therefore, it is worthwhile to study how to reasonably calculate the economic benefits during the life cycle of CSES while ensuring the calculation efficiency.

For the methods of quantifying the capacity decay of CSES during operation, quantifying the capacity decay in the time scale of only one day [15] cannot reflect the real operation of the energy storage system, because the capacity decay is a slow process. Quantifying the capacity decay of the energy storage system accurately each year by the rain flow method [16] will lead to increased computing scale and a difficult solution when the number of community users increases. In addition, the power loss cost factor [17] did not consider the change during the life cycle of the energy storage system. Therefore, how quantifying the capacity decay of CSES during operation to reflect the real operation is another important issue.

For the solving method of the model, the heuristic algorithm combined with the CPLEX solver [18,19] cannot guarantee the optimality of the results, and even in most cases, it cannot calculate the approximation between the results and the optimal solution. The iterative method [16] needs to solve the constructed two-layer model in each iteration, which is of low efficiency. Therefore, how to improve the solving efficiency of the model while ensuring the quality of the solution is also a problem that needs to be solved.

1.3. Contributions and Paper Organization

For the above problems, this paper proposes a CSES sizing and configuration method based on a cooperative game. First, a cooperative sharing mode of CSES is proposed as the basis for subsequent model construction. Second, a CSES sizing and configuration model is established. Community users and energy storage operators form an alliance to play a cooperative game with the goal of minimizing the annual electricity cost during the life cycle of CSES, and a power loss cost factor is proposed to describe the loss due to the capacity decay of energy storage. Then, a distributed and cooperating solving method based on ADMM is used to realize the distributed and cooperating computation among community users, and the cost-sharing is based on the bilateral Shapley value method. Finally, the feasibility of the proposed model and the superiority of CSES compared with different energy storage configuration modes are verified by simulation. The comparison of the work of this paper with other works in the literature review is shown in

Table 1. The comparison of the work of this paper with other works.

Literature	Duration of Cost Calculation	CSES’s Capacity Decay Quantifying Method	Solving Method
Lu et al. [13]	A single typical day	No consideration	Genetic Algorithms
Edward Barbour et al. [14]	A month	No consideration	No description
Cheng et al. [15]	Multiple typical days of a year	Capacity loss for one day	MILP solver
Xie et al. [16]	Several typical days of each year	Rain flow method	iterative method
Chen et al. [17]	A single typical day	Static power loss cost factor	ADMM
Shuai et al. [18,19]	A single typical day	No considerations	Heuristic algorithm combined with the CPLEX solver
This paper	Several typical days of each year	Dynamic power loss cost factor	A distributed and cooperating solving method based on ADMM

.

The main innovation points of this paper are as follows.

(1)

A CSES configuration framework that considers the decisions of multiple subjects is proposed, which could address the subjective uncertainty caused by the decision preferences of each subject through the cooperative game and maximize the total benefits of the community.

(2)

A dynamic power loss cost factor that increases year by year over the life cycle of CSES is proposed, which quantifies the capacity decay of CSES by power loss cost and regards the capacity of CSES as fixed. It could address the difficulty in calculating the loss caused by the CSES’s capacity decay with high calculating efficiency.

(3)

A distributed and cooperating solution method based on the ADMM algorithm is proposed, which transforms transactions among multiple subjects into transactions between two subjects by equating the multi-coupling constraints with the double-coupling constraints. It could improve the model’s low-solving efficiency when there are many community users.

The rest of this paper is structured as shown below. Chapter 2 introduces the sizing and configuration framework of CSES, including the cooperative sharing mode of CSES, the structure of the sizing and configuration model, and the modeling of uncertainty. Chapter 3 presents the sizing and configuration model of CSES based on a cooperative game and cost-sharing model. The model-solving method is proposed in Chapter 4. Chapter 5 conducts numerical experiments on various cases to verify the feasibility and validity of the model. Finally, conclusions and future research directions are given in Chapter 6.

2. Framework of CSES Configuration

This section introduces the sizing and configuration framework of CSES. Specifically, firstly a cooperation and sharing model with multiple transaction types is proposed for shared energy storage communities. Then, the cooperation game framework in the sizing and configuration model is introduced. Finally, the uncertainties involved in the model are modeled, including objective uncertainty modeling and subjective uncertainty modeling.

2.1. CSES Cooperation and Sharing Model

The CSES studied in this paper is jointly invested by all community users and managed by energy storage operators. To maximize the economic benefits brought by CSES, this paper designs a cooperation and sharing model of CSES based on the user’s aggregation in the same community according to the concept of sharing economy, as shown in Figure 1, which could integrate the demand complementarity of different users, promote renewable energy consumption, reduce carbon emissions, ultimately reduce user costs and improve the overall load shape of the community.

As shown in Figure 1, the 4 transaction types involved in a CSES community are as follows.

①

Electricity sharing among different community users. Roof-top PV users no longer adopt “all injected into power grid” mode or “local utilization and surplus injected into power grid” mode but share the surplus PV power with other community users who need electricity, which could reduce the electricity purchased by the community from the grid and the overall electricity cost of the community.

②

Electricity transactions between CSES and community users. Roof-top PV users could store surplus PV power in CSES and release them when the PV power is low.

③

Electricity transactions between community users and the grid. When PV power and energy storage discharge power cannot meet the power demand of community users, they need to purchase electricity from the grid to meet their demands. Similarly, during the peak period of PV power, PV users can also sell electricity to the grid for profit.

④

Electricity transactions between CSES and the grid. Generally speaking, CSES will not sell electricity to the grid due to the charging and discharging losses. CSES could buy electricity at low prices during low load valleys and discharge it at peak load times to reduce the peak electricity purchased by the community.

Ultimately, through the CSES cooperation and sharing model as shown in Figure 1, in each period, PV users with surplus power share it with other users in need or store it in CSES and sell the rest to the grid. Because sharing electricity among community users saves more money for the whole community compared with trading with the grid. In addition, as consumers, ordinary users could obtain power directly from PV users or CSES, realizing the CSES sharing among users. CSES could promote the surplus PV power consumption and release it when they are in demand, and it can also purchase power for storage at low valley tariffs and release it at peak tariffs, ultimately reducing the users’ costs.

2.2. CSES Sizing and Configuration Model Based on Cooperative Game

Compared with the traditional “all injected into power grid” mode or “local utilization and surplus injected into power grid” mode, CSES makes the operation mode of the whole community complicated and diversified, and the different operation modes will directly affect the electricity cost of each user and the overall electricity cost of the community, thus affecting the sizing and configuration of CSES.

Given the complicated and diversified operation modes of the community, this paper builds a CSES sizing and configuration model based on a cooperative game, and the framework of the cooperation game is shown in Figure 2.

A cooperative game is a positive-sum game in which the interests of all subjects in the game are increased, or at least the interests of some subjects are increased while the interests of other subjects are not harmed, and thus the interests of the whole community are increased [20]. In the cooperative game, all the users and energy storage operators in the community form an alliance and reach a consensus on the interactive power and the CSES configuration capacity according to the bargaining theory and determine the CSES configuration capacity and the optimal operation mode of the community to maximize the overall alliance benefits. Then, through a fair cost-sharing model, the electricity cost of each subject is lower than before to maintain the stability of the alliance.

2.3. Uncertainty Modeling

The uncertainty of CSES operation mainly comes from two aspects. One is the objective uncertainty, which is the source and load uncertainties caused by the available output of roof-top PV and the variable power demand of users. The other is subjective uncertainty, which is manifested in the behavior uncertainty of every trading subject [21].

Obviously, the former is caused by the combination of the external environment, weather factors, and the consumption behavior of community users, which could be estimated based on a large amount of existing data. Considering the real-time subjective preferences, the latter is a kind of adaptive user behavior to maximize their interests, which is closely related to the community operation and the behavior of other users.

The kernel k-means clustering (KKM) algorithm [22,23] is one of the classical algorithms in kernel clustering. As a nonlinear extension of the k-means clustering algorithm, KKM could transform the nonlinear indivisible problem of data in the original space to the linear separable problem in high-dimensional space, improving the accuracy of clustering results. Therefore, for the objective source and load uncertainty, this paper uses the kernel k-means clustering (KKM) algorithm based on the Laplace kernel to obtain multiple typical daily scenarios and their distribution probabilities based on regional PV and load yearly data. As a widely used kernel function, the Laplace kernel has a low sensitivity to the kernel width. The number of clusters is 4 and the kernel width parameter is 10.

For the subjective uncertainty related to the subjective preferences of the trading subject, the cooperative game model in Section 2.2 simulates the profit-seeking behavior of each trading subject. All the users and energy storage operators in the community form an alliance to minimize the overall community electricity cost. At the same time, the cost-sharing model built in Section 3.2 gives each trading subject deserved benefit to ensure the fairness of the transaction and the stability of the alliance.

3. Mathematic Formulations

This chapter mainly includes two parts: CSES sizing and configuration model building and the cost-sharing model building based on the bilateral Shapley method.

3.1. CSES Sizing and Configuration Model

CSES sizing and configuration model is as follows, in which the decision variables are the CSES capacity E_norm, purchasing and selling electricity of users and CSES, and power interaction among users and CSES. Equations (1)–(6) are the target function minimizing the total annual cost of the community during the life cycle of CSES. Equations (7)–(12) are CSES constraints and Equations (13)–(15) are power balance constraints. Total annual cost C_total equals annual CSES investment and maintenance cost C_est plus annual operation cost C_ope. C_ope equals annual electricity purchasing cost minus annual electricity selling income plus annual power loss cost.

Target function:

$$\min C_{\mathrm{total}}=\min C_{\mathrm{est}}+C_{\mathrm{ope}},$$

(1)

$$C_{\mathrm{est}}=\beta E_{\mathrm{norm}}{c}_{\mathrm{bat}}+E_{\mathrm{norm}}{c}_{\mathrm{o}\&\mathrm{m}},$$

(2)

$$\beta =\frac{\alpha \cdot {(1+\alpha )}^{Year}}{{(1+\alpha )}^{Year}-1},$$

(3)

$$C_{\mathrm{ope}}=\frac{1}{Year}{\displaystyle \sum _{k=1}^{Year}365\cdot E_{\mathrm{w}}({\displaystyle \sum _{i=1}^N{C}_i^{\mathrm{ope}}(k,w)})+\frac{1}{Year}{\displaystyle \sum _{k=1}^{Year}365\cdot E_{\mathrm{w}}(C_{\mathrm{loss}}(k,w))}},$$

(4)

$$C_i^{\mathrm{ope}}(k,w)={\displaystyle \sum _{t=1}^{24}{\lambda }_t^{\mathrm{b}}{P}_{i,t}^{\mathrm{b}}(k,w)-{\lambda }_t^{\mathrm{s}}{P}_{i,t}^{\mathrm{s}}(k,w)},$$

(5)

$$C_{\mathrm{loss}}(k,w)={\displaystyle \sum _{t=1}^{24}{c}_{\mathrm{e}}(k)\cdot (P_t^{\mathrm{c}}(k,w)+P_t^{\mathrm{d}}(k,w))},$$

(6)

where β is the annualizing operator for the calculation of annual CSES investment and maintenance cost, which is defined in Equation (3); α is annual interest rate; Year is the life span of energy storage batteries; c_bat is the unit capacity investment cost of energy storage; c_o&m is the unit capacity maintenance cost of energy storage; E_W(·) means expectations value in multiple scenarios; C_i^ope(k,w) is the daily operation cost of user i under scenario w in year k, as shown in Equation (5); λ_t^b and λ_t^s are the purchasing and selling price of electricity, respectively, at time t; P_i,t^b(k,w) and P_i,t^s(k,w) are the purchasing and selling electricity of user i, respectively, at time t under scenario w in year k; C_loss(k,w) is the daily power loss cost of energy storage under scenario w in year k, which is defined in Equation (6); c_e(k) is the power loss cost factor in year k, which gradually increases with the operation time of energy storage; P_t^c(k,w) and P_t^d(k,w) are the charge and discharge power of energy storage, respectively, at time t under scenario w in year k; N is the total number of community users.

Constraints:

CSES constraints:

$$E_{t+1}(k,w)=E_t(k,w)+P_t^{\mathrm{c}}(k,w){\eta }_{\mathrm{c}}-\frac{1}{{\eta }_{\mathrm{d}}}{P}_t^{\mathrm{d}}(k,w),$$

(7)

$$0\le P_t^{\mathrm{c}}(k,w)\le a_t(k,w)r{E}_{\mathrm{norm}},$$

(8)

$$0\le P_t^{\mathrm{d}}(k,w)\le b_t(k,w)r{E}_{\mathrm{norm}},$$

(9)

$$0\le a_t(k,w)+b_t(k,w)\le 1,$$

(10)

$$E_0(k,w)=E_{24}(k,w),$$

(11)

$$E_{\mathrm{norm}}(1-D)\le E_t(k,w)\le E_{\mathrm{norm}},$$

(12)

Power balance constraints:

$$\begin{array}{l}{P}_{i,t}^{\mathrm{b}}(k,w)+d_{i,t}(k,w)-P_{i,t}^{\mathrm{s}}(k,w)-c_{i,t}(k,w)\\ -{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^N{P}_{ij,t}(k,w)}=l_{i,t}(k,w)-{\mathrm{g}}_{i,t}(k,w),\end{array}$$

(13)

$$P_t^{\mathrm{c}}(k,w)={\displaystyle \sum _{i=1}^N{c}_{i,t}(k,w)}+P_t^{\mathrm{b},\mathrm{ESS}}(k,w)$$

(14)

$$P_t^{\mathrm{d}}(k,w)={\displaystyle \sum _{i=1}^N{d}_{i,t}(k,w)}+P_t^{\mathrm{s},\mathrm{ESS}}(k,w)$$

(15)

where E_t(k,w) is the electricity storage at time t under scenario w in year k; η_c and η_d are the charging and discharging efficiency of energy storage, respectively; r is the maximum charge or discharge rate of energy storage; a_t(k,w) and b_t(k,w) are binary variables, ensuring that energy storage cannot be charged and discharged simultaneously; D is the maximum discharge depth; c_i,t(k,w) and d_i,t(k,w) are the charging and discharging power between energy storage and user i at time t under scenario w in year k; P_ij,t(k,w) is the interactive power between user i and user j at time t under scenario w in year k; l_i,t(k,w) is the power demand of user i at time t under scenario w in year k; g_i,t(k,w) is the PV predicted power of user i at time t under scenario w in year k, which is always 0 for ordinary users; P_t^b,ESS(k,w) and P_t^s,ESS(k,w) are the purchasing and selling electricity of energy storage at time t under scenario w in year k.

3.2. Cost-Sharing Model Based on Bilateral Shapley Value

The key to alliance cooperation is the fairness of interest distribution. The existing cost-sharing methods mainly include the Nash solution, Shapley value method, nucleolus solution, non-separable cost gap method, etc. [17] While as a common cost-sharing method in the cooperative game, Shapley value method faces the problem of the sub-alliance explosion, which will result in a huge amount of calculation [24]. With reduced computing time, the bilateral Shapley value method could be treated as an approximate solution of the Shapley value method in engineering [25].

The principle of the Shapley value method is to allocate the total alliance cost according to the marginal expected cost brought by each user, and the cost to be paid by user i is defined in Equation (16).

$${\phi }_i={\displaystyle \sum _{S\subseteq S_i}\pi \left(\left|S\right|\right)}\left[v\left(S\right)-v\left(S\backslash i\right)\right]+v\left(\left\{i\right\}\right),$$

(16)

where φ_i is the cost allocated to user i; S_i is the set of all alliances containing user i; |S| is the number of members contained in subset S; π(|S|) is the probability of containing S in all possible alliances; v(S) − v(S\i) is the marginal cost contribution of user i to alliance S; v({i}) is the original electricity cost of user i; v is the characteristic function, which is specifically the annual electricity cost function of cooperative alliance in this paper; S\i is the alliance remaining in S after removing user i.

The calculated expression of π(|S|) is as Equation (17):

$$\pi \left(\left|S\right|\right)=\frac{(N-\left|S\right|)!(\left|S\right|-1)!}{N!}.$$

(17)

According to Equation (17), the time complexity of the regular Shapley value method depends on the factorial of participants number, which is O(n!). To avoid the problem of dimensional disaster caused by a large number of users, this section uses the bilateral Shapely value method [26] for cost sharing.

The bilateral Shapley value method is to divide the participants of cooperative alliance M into only two subjects {i} and {M\i} when calculating the cost of user i. By sacrificing some unnecessary precision, the computational complexity is reduced. Therefore, the cost to be paid by user i is defined in Equation (18):

$${\phi }_i^{\mathrm{new}}=\frac{1}{2}(v(\mathrm{M})-v(\mathrm{M}\backslash i)+v(\{i\})),$$

(18)

where φ_i^new is the cost allocated to user i; v(M) − v(M\i) is the marginal cost contribution of user i to alliance M; v({i}) is the original annual cost of user i.

Generally, Equation (18) cannot ensure the adequate allocation of community electricity costs, the rest of which could be calculated as Equation (19):

$$\Delta v=v\left(\mathrm{M}\right)-{\displaystyle \sum _{i\in M}{\phi }_i^{\mathrm{new}}},$$

(19)

where Δv is the rest of the community electricity costs.

Therefore, to ensure reasonableness and fairness, Δv is further proportionally allocated as Equation (20):

$${\phi }_i^{\mathrm{final}}={\phi }_i^{\mathrm{new}}+\frac{{\phi }_i^{\mathrm{new}}}{{\displaystyle \sum _{i\in M}{\phi }_i^{\mathrm{new}}}}\Delta v,$$

(20)

where φ_i^final is the modified cost allocated to user i.

4. Solving Algorithm

Since the total annual cost during the life cycle of CSES is considered in the model, the computing scale will increase rapidly as the number of users increases. Therefore, in order to improve computational efficiency and shorten the solving time, this paper adopts a distributed and cooperating solving method based on ADMM [27,28,29] to achieve distributed and cooperating computation among community users.

Since the existing CSES sizing and configuration model is a mixed integer programming problem, which is difficult to solve directly by the ADMM algorithm, the integer constraints in Equations (8)–(10) are linearized as shown in Equations (21) and (22).

The linearization actually ignores the integer variables characterizing the charging and discharging states of CSES. This is because the presence of energy storage charging and discharging losses makes it possible to ignore the integer variables in the case of the optimal solution since the energy storage will not be charged and discharged simultaneously even without the limitation of the integer variables.

$$0\le P_t^{\mathrm{c}}(k,w)\le r{E}_{\mathrm{norm}},$$

(21)

$$0\le P_t^{\mathrm{d}}(k,w)\le r{E}_{\mathrm{norm}}.$$

(22)

The linearized CSES sizing and configuration model can be expressed as shown in Equation (23). Model (23) is a linear programming problem, so it is also a convex optimization problem, which can ensure the convergence of ADMM [30].

$$\left\{\begin{array}{l}\min C_{\mathrm{total}}\\ s.t.\mathrm{Eqs}.(7),(11)-(15),(21)-(22)\end{array}\right..$$

(23)

Since the CSES constraints and the power balance constraints are multi-coupled between users and CSES, the auxiliary variables P_j-i,t, c_E-i,t, d_E-i,t, and E^’_norm are introduced and the double coupling constraints as shown in Equations (24)–(26) is added to equate the multi-coupling constraints [31].

$$P_{i-j,t}+P_{j-i,t}=0,\forall i,j,$$

(24)

$$c_{i-E,t}-d_{i-E,t}-c_{E-i,t}+d_{E-i,t}=0,\forall i,$$

(25)

$$E_{\mathrm{norm}}-E_{\mathrm{norm}}^{\prime }=0,$$

(26)

where P_j-i,t is the interactive power that user j expects to trade with user i; c_E-i,t is the electricity that CSES expects to receive from user i; d_E-i,t is the electricity that CSES expects to discharge to user i; E^’_norm is the expected configuration capacity of CSES.

When the constraints in Equations (24)–(26) are satisfied, trading consensuses are reached among each subject. After completing the decoupling transformation, the specific steps for the distributed solution of problem (23) based on the ADMM algorithm are as follows.

Build the decision model of each subject.

Community decision model:

$$\min L_{\mathrm{group}}=\min C_{\mathrm{est}}+{\lambda }_{\mathrm{est}}(E_{\mathrm{norm}}-E_{\mathrm{norm}}^{\prime })+\frac{\rho }{2}{‖E_{\mathrm{norm}}-E_{\mathrm{norm}}^{\prime }‖}_2^2.$$

(27)

User decision model:

$$\begin{array}{l}\min L_i=\min C_i^{\mathrm{ope}}+{\displaystyle \sum _j^{}{\displaystyle \sum _{t=1}^T{\lambda }_{i-j,t}(P_{i-j,t}+P_{j-i,t})}+\frac{\rho }{2}}{\displaystyle \sum _j^{}{{\displaystyle \sum _{t=1}^T‖P_{i-j,t}+P_{j-i,t}‖}}_2^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{t=1}^T{\lambda }_{i-E,t}(c_{i-E,t}-d_{i-E,t}-c_{E-i,t}+d_{E-i,t})}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\rho }{2}{{\displaystyle \sum _{t=1}^T‖c_{i-E,t}-d_{i-E,t}-c_{E-i,t}+d_{E-i,t}‖}}_2^2.\end{array}$$

(28)

CSES decision model:

$$\begin{array}{l}\min L_{\mathrm{E}}=\min C_{\mathrm{loss}}+{\displaystyle \sum _i{\displaystyle \sum _{t=1}^T{\lambda }_{E-i,t}(c_{E-i,t}-d_{E-i,t}-c_{i-E,t}+d_{i-E,t})}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\rho }{2}{\displaystyle \sum _i{{\displaystyle \sum _{t=1}^T‖c_{E-i,t}-d_{E-i,t}-c_{i-E,t}+d_{i-E,t}‖}}_2^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\lambda }_{\mathrm{est}}^{\prime }(E_{\mathrm{norm}}^{\prime }-E_{\mathrm{norm}})+\frac{\rho }{2}{‖E_{\mathrm{norm}}^{\prime }-E_{\mathrm{norm}}‖}_2^2,\end{array}$$

(29)

where λ_est, λ_est^’, λ_i-j,t, λ_i-E,t and λ_E-i,t are Lagrange multipliers; ρ is the penalty factor, whose value is 5 × 10⁻⁴.

Each subject updates his own decision through local computing, and only electricity trading information and CSES capacity information are exchanged among the subjects. Let k denote the number of iterations, and in each iteration, the following steps are performed.

The community updates its decision E_norm(k + 1) through Equation (30) and passes the updated decision information to other subjects.

$$E_{\mathrm{norm}}(k+1)=\arg \min L_{\mathrm{group}}.$$

(30)

Each user updates its decision P_i-j(k + 1), c_i-E(k + 1), and d_i-E(k + 1) through Equation (31) and passes the updated decision information to other subjects.

$$\left(P_{i-j}(k+1),c_{i-E}(k+1),d_{i-E}(k+1)\right)=\arg \min L_i.$$

(31)

CSES updates its decision E_norm^’(k + 1), c_E-i(k + 1), and d_E-i(k + 1) through Equation (32) and passes the updated decision information to other subjects.

$$\left(E_{\mathrm{norm}}^{\prime }(k+1),c_{E-i}(k+1),d_{E-i}(k+1)\right)=\arg \min L_{\mathrm{E}}.$$

(32)

After one round of iterations, the Lagrange multiplier is updated according to Equations (33)–(37).

$${\lambda }_{\mathrm{est}}(k+1)={\lambda }_{\mathrm{est}}(k)+\rho (E_{\mathrm{norm}}-E_{\mathrm{norm}}^{\prime }),$$

(33)

$${\lambda }_{i-j,t}(k+1)={\lambda }_{i-j,t}(k)+\rho (P_{i-j,t}+P_{j-i,t}),$$

(34)

$${\lambda }_{i-E,t}(k+1)={\lambda }_{i-E,t}(k)+\rho (c_{i-E,t}-d_{i-E,t}-c_{E-i,t}+d_{E-i,t}),$$

(35)

$${\lambda }_{E-i,t}(k+1)={\lambda }_{E-i,t}(k)+\rho (c_{E-i,t}-d_{E-i,t}-c_{i-E,t}+d_{i-E,t}),$$

(36)

$${\lambda }_{\mathrm{est}}^{\prime }(k+1)={\lambda }_{\mathrm{est}}^{\prime }(k)+\rho (E_{\mathrm{norm}}^{\prime }-E_{\mathrm{norm}}).$$

(37)

Update the number of iterations: k = k + 1.

Determine the convergence of ADMM.

The imbalance of the model solution △ is shown in Equations (38) and (39). △ is used to quantify the degree of transaction consensuses among subjects in the kth iteration. △= 0 means all subjects have reached trading consensuses completely.

$$\Delta (k)={\displaystyle \sum _i{\Delta }_i}(k)+(E_{\mathrm{norm}}-E_{\mathrm{norm}}^{\prime }),$$

(38)

$${\Delta }_i(k)={\displaystyle \sum _t\left[{\displaystyle \sum _{j\ne i}(P_{i-j,t}+P_{j-i,t})}+(c_{i-E,t}-d_{i-E,t}-c_{E-i,t}+d_{E-i,t})\right]}.$$

(39)

If Equation (40) is satisfied then the iteration is terminated, otherwise, return to step 2) and enter the next iteration until the convergence condition is satisfied or the maximum iterations are reached.

$$\left\{\begin{array}{l}\Delta (k)-\Delta (k-1)\le \delta ,\mathrm{the} k\mathrm{th} \mathrm{iteration} \mathrm{converge}\\ k>k_{\max },\mathrm{the} \mathrm{algorithm} \mathrm{does} \mathrm{not} \mathrm{converge}\end{array}\right.$$

(40)

5. Case Study

This chapter focuses on the design, calculation, and analysis of the cases from five aspects: the analysis of the sizing and configuration results and operation strategy, the comparison of different energy storage configuration modes, the influence of the proportion of PV users on the energy storage configuration capacity, the sensitivity analysis for the electricity price and feed-in tariff, and the influence of uncertainty modeling on the energy storage configuration capacity. The main parameters involved in the case are shown below.

Load and distributed PV parameters. The industrial community in this case contains eight ordinary users and twelve PV users, which means the proportion of PV users is 60%. Four typical scenarios and their occurrence probabilities are generated based on the historical data of community load and PV power taken from the Open Power System Data platform [32].

Other key parameters. Other key parameters of the case are shown in

Table 2. Key parameters.

Parameter	Unit	Value
α	——	0.01
Year	——	15
cbat	CNY/kW	2000
co&m	CNY/kW	100
ce0	——	0.01
ηc	%	95
ηd	%	95
r	——	0.9
D	——	0.95
λhighb	CNY/kWh	1.1
λmidb	CNY/kWh	0.6
λlowb	CNY/kWh	0.3
λts	CNY/kWh	0.35

, which mainly include the electricity price, feed-in tariff, annual interest rate, the performance and cost parameters of energy storage, etc.

For the electricity price, there are three periods of the day: peak hours (10:00–15:00, 18:00–21:00), flat hours (7:00–10:00, 15:00–18:00, 21:00–24:00), and valley hours (24:00–7:00).

The feed-in tariff λ_t^s take the value according to the electricity price of local desulfurized coal.

The power loss cost factor of CSES increases year by year over the life cycle with an annual growth rate of 10%.

5.1. Sizing and Configuration Results and Operation Strategy Analysis

To highlight the benefits brought by CSES, this paper conducts simulations for the cases before and after CSES configuration, and the data and parameter settings in the cases are the same. The results of both cases are shown in

Table 3. Simulation results before and after CSES configuration.

Attributes	After CSES Configuration	Before CSES Configuration
Energy storage capacity/kWh	806.81	0
Annual CSES investment and maintenance cost/CNY	197,100	0
Annual operation cost/CNY	660,900	893,000
Total annual cost/CNY	858,000	893,000

.

Table 3 shows that the configuration of CSES can save CNY 35,000 in total annual cost and improve the economy by 3.92%, and the community can save CNY 232,100 in annual operation cost and improve the economy by 25.6%. It shows that by investing in CSES, the scale effect can be exploited to improve economic benefits and reduce the total annual cost effectively. In addition, in the view of the grid, the configuration of CSES not only relieves the grid’s peaking pressure but also promotes the consumption of new energy and reduces carbon emissions, with both economic and environmental benefits.

After determining the total annual cost, each user shares the cost according to the cost-sharing model constructed in Section 3.2. The annual cost-sharing results of some users are shown in

Table 4. Cost-sharing results.

Users	Original Total Annual Cost/CNY	Current Total Annual Cost/CNY	Economy Improvement
User 1	61,800	60,300	2.43%
User 2	62,900	59,900	4.77%
User 3	25,800	23,900	7.36%
User 4	27,100	25,000	7.75%
User 5	38,500	37,000	3.9%

.

The original total annual cost of user I is defined in Equations (41)–(44).

$$C_i^{\mathrm{ori}}=\frac{1}{Year}{\displaystyle \sum _{k=1}^{Year}365\cdot E_{\mathrm{w}}(C_i^{\mathrm{ori}}(k,w))},$$

(41)

$$C_i^{\mathrm{ori}}(k,w)={\displaystyle \sum _{t=1}^{24}{\lambda }_t^{\mathrm{b}}{P}_{i,t}^{\mathrm{b}\_\mathrm{ori}}(k,w)-{\lambda }_t^{\mathrm{s}}{P}_{i,t}^{\mathrm{s}\_\mathrm{ori}}(k,w)},$$

(42)

$$P_{i,t}^{\mathrm{b}\_\mathrm{ori}}(k,w)=\left\{\begin{array}{l}{l}_{i,t}(k,w)-{\mathrm{g}}_{i,t}(k,w),l_{i,t}(k,w)-{\mathrm{g}}_{i,t}(k,w)>0,\\ 0,l_{i,t}(k,w)-{\mathrm{g}}_{i,t}(k,w)\le 0,\end{array}\right.$$

(43)

$$P_{i,t}^{\mathrm{s}\_\mathrm{ori}}(k,w)=\left\{\begin{array}{l}0,l_{i,t}(k,w)-{\mathrm{g}}_{i,t}(k,w)>0,\\ {\mathrm{g}}_{i,t}(k,w)-l_{i,t}(k,w),l_{i,t}(k,w)-{\mathrm{g}}_{i,t}(k,w)\le 0,\end{array}\right.$$

(44)

where C_i^ori is the original total annual cost of user i; C_i^ori(k,w) is the original daily cost of user i under scenario w in year k as shown in Equation (42); P_i,t^b_ori(k,w) and P_i,t^s_ori(k,w) are the original purchasing and selling electricity of user i, respectively, at time t under scenario w in year k as shown in Equations (43) and (44).

Each participant has the incentive to join the alliance only if the alliance satisfies the group rationality condition (the alliance benefits are greater than the sum of the original benefits of individuals) and the individual rationality condition (the benefits of each individual in the alliance are greater than before) [33,34]. As shown in Table 4, user 1 and user 2 are ordinary users, and user 3~user 5 are PV users, the annual cost of each user reduces in different degrees, and the reduction ranges from 2.43% to 7.75%. Generally, PV users tend to share more profits because they shared surplus power with other users in need or store it in CSES and sell the rest to the grid, making more contributions to the community compared to the ordinary users. As a result, they will have a lower marginal expected cost. Additionally, ordinary users naturally tend to share fewer profits for the received power from PV users and CSES. Finally, each user gains more benefits in the cooperation, which satisfies the individual rationality condition, so each user is motivated to participate in the cooperation.

To further analyze the community’s operation after configuring CSES, the community operation strategy during a certain day with the optimal CSES configuration capacity is obtained as shown in Figure 3 based on the simulation results. Where the net load is equal to the total community load minus the total PV power.

In Figure 3, the CSES initial energy is the lowest value under the limit of the maximum discharge depth. Between 1:00 and 7:00, the PV power is small, the community mainly purchases electricity from the grid to meet the electricity demand, and at this time the time-of-use price is in the valley hours, CSES purchases cheap electricity from the grid for storage. When the time-of-use price is in the flat hours at 8:00, CSES discharges to reduce the community purchasing electricity from the grid. After 9:00, the PV power becomes greater than the community electricity demand. CSES uses the excess PV power to charge, and the rest is sold to the grid. After 16:00, the PV power gradually becomes less than the community power demand. However, because the time-of-use price is in the flat hours at this time, CSES does not discharge, and the community purchases electricity from the grid. After 18:00, when the time-of-use price is in the peak hours, CSES begins to discharge to reduce the electricity purchased from the grid, maximizing the benefits of CSES.

5.2. Comparison of Different Energy Storage Configuration Modes

To further highlight the benefits brought by the CSES, this paper also conducts simulations for the cases with CSES and private user energy storage. The data and parameter settings in the cases are the same. The results of both cases are shown in

Table 5. Simulation results for CSES and private user energy storage.

Attributes	CSES	Private User Energy Storage
Energy storage capacity/kWh	806.81	1048.8
Annual CSES investment and maintenance cost/CNY	197,100	256,200
Annual operation cost/CNY	660,900	609,000
Total annual cost/CNY	858,000	865,200

.

Table 5 shows that the capacity of private user energy storage is 29.99% greater than CSES, which leads to lower annual operation costs compared with CSES mode. However, because of the low utilization rate of energy storage, the saved annual operation cost of private user energy storage mode is less than the increased investment and maintenance cost compared with CSES mode. As a result, the total annual cost of the private energy storage mode is CNY 7,200 more than CSES mode, which shows the superiority of CSES clearly.

5.3. Influence of the Proportion of PV Users on the CSES Capacity

This section adjusts the proportion of PV users and calculates the CSES configuring capacity and total annual cost with different proportions of PV users. The results are shown in

Table 6. Simulation results with different proportions of PV users.

Proportion of PV Users	CSES Capacity/kWh	Original Total Annual Cost/CNY	Current Total Annual Cost/CNY	Economy Improvement
80%	925.69	749,200	784,300	4.48%
60%	806.81	858,000	893,000	3.92%
40%	736.22	1,015,200	1,037,800	2.18%

.

The original total annual cost of the community C^ori is defined in Equation (45).

$$C^{\mathrm{ori}}={\displaystyle \sum _{i=1}^N{C}_i^{\mathrm{ori}}},$$

(45)

Table 6 shows that when the proportion of PV users increases from 60% to 80%, the CSES configuration capacity increases by about 120 kWh, and the economic improvement increases from 3.92% to 4.48%. Within a certain range, the higher the proportion of PV users, the more capacity of CSES is configured, which improves the economy more significantly. This is mainly due to the fact that the utilization potential of PV and the demand for energy storage increase with the proportion of PV users. Energy storage can store more of the surplus PV power and discharge it during the peak hours to reduce the annual operation cost by reducing the electricity purchased at the peak price.

5.4. Sensitivity Analysis

To analyze the impact of the electricity price and feed-in tariff on the CSES configuration capacity, five cases are designed as shown in

Table 7. Cases parameters.

Cases	Peak Price λhighb (CNY/kWh)	Flat Price λmidb (CNY/kWh)	Valley Price λlowb (CNY/kWh)	Feed-in Tariff λts (CNY/kWh)
Case 1	1.1	0.6	0.3	0.2
Case 2	1.2	0.7	0.4	0.2
Case 3	1.3	0.8	0.5	0.2
Case 4	1.1	0.6	0.3	0.15
Case 5	1.1	0.6	0.3	0.1

, where the parameters of case 1 are the same as Section 5.1 and are just used for control.

Calculate the CSES configuring capacity and the total annual cost of five different cases, and the results are shown in

Table 8. Simulation results for different cases.

Cases	CSES Capacity/kWh	Original Total Annual Cost/CNY	Current Total Annual Cost/CNY	Economy Improvement
Case 1	806.81	858,000	893,000	3.92%
Case 2	821.59	996,200	1,052,800	5.38%
Case 3	854.98	1,134,100	1,212,600	6.47%
Case 4	814.13	858,400	906,100	5.26%
Case 5	840.06	858,700	919,100	6.57%

.

Comparing the results of case 1, case 2, and case 3, when the electricity price increases gradually, the total annual cost increases as well. Specifically, when the electricity price increases by CNY 0.1, the CSES configuration capacity increases by approximately 15 kWh, and the economic improvement increases from 3.92% to 5.38%. This is mainly due to the fact that CSES makes the community purchase less electricity from the grid, so when the electricity price increases, CSES can further exploit its advantages, thus creating greater economic benefits.

Comparing the results of case 1, case 4, and case 5, when the feed-in tariff decreases gradually the total annual cost increases. Specifically, when the feed-in tariff decreases by CNY 0.1, the CSES configuration capacity increases by about 35 kWh, and the economic improvement increases from 3.92% to 6.57%. This is mainly due to the fact that the community will gain fewer benefits from selling surplus PV power to the grid when the feed-in tariff decreases. Therefore, it tends to share surplus power among users in need or store surplus power into CSES, which means more CSES is needed, and it could save more cost for the community.

In summary, the CSES configuration capacity will inevitably be affected by the electricity price and feed-in tariff, and the sensitivity to the electricity price and feed-in tariff is high. So, both must be taken strictly in accordance with the local electricity price and feed-in tariff when sizing and configuring, otherwise, the CSES configuration capacity will be directly affected.

5.5. Influence of Uncertainty Modeling on CSES Capacity

To study the impact of uncertainty modeling on CSES capacity, this section compares the differences in CSES configuration capacity of the multi-scenario case with the four single-scenario cases as shown in

Table 9. Simulation results in different scenario cases.

Attributes	Scenario 1	Scenario 2	Scenario 3	Scenario 4	Multi-Scenario
CSES capacity/kWh	736.44	884.06	806.16	736.37	806.81

. In the multi-scenario case, the randomness of PV power and users’ demand is taken into account, which is specifically represented by several typical daily scenarios, and in the single-scenario case, the CSES configuration capacity is based on the PV power and demand data of only one scenario.

Table 9 shows that the CSES configuration capacity of scenario 2 is the largest and the CSES configuration capacity of scenario 4 is the smallest. The CSES configuration capacity of the multi-scenario is in between. If the configuration is carried out according to scenario 2, it will lead to high investment costs and low utilization of energy storage. If the configuration is carried out according to scenario 4, it will lead to the reduced operational life of CSES due to excessive discharging. The multi-scenario case integrates the impact of each scenario and minimizes the electricity cost over a long-time scale, making it a more economical choice.

6. Conclusions

Sizing and configuring CSES according to the actual demand of community users is important for the popularization of CSES and the development of user-side energy storage. This paper proposes a CSES sizing and configuration model based on a cooperative game, in which a dynamic power loss cost factor is used to quantify the capacity decay of CSES. Based on this, a distributed and cooperating solving method based on ADMM is used to realize the distributed and cooperating computation among community users to improve the solution efficiency, and the bilateral Shapley value method is used to allocate the total alliance cost according to the marginal expected cost brought by each user to improve the stability of the alliance. Finally, through simulation analysis, the following conclusions can be obtained.

The proposed model can be used for the sizing and configuration of community energy storage under the sharing model. On the one hand, the configuration of CSES reduces the total annual cost of the community to a certain extent, while every member within the coalition gains more benefits as well. On the other hand, the utilization of the sharing model also helps to improve the penetration of solar energy in a more efficient way. As a result, it could avoid high investment costs, low utilization of energy storage, and reduced operational life of CSES.

However, it is important to note that this paper only considers the sharing of CSES among the community. In fact, CSES can also be shared among multiple neighboring communities or profited by providing auxiliary services to the grid, and its economic and social benefits will be further enhanced, which is the focus of the subsequent work in this paper.